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Complex variables conformal mapping trig identity

  1. Apr 19, 2010 #1
    1. The problem statement, all variables and given/known data
    map the function [tex] \begin{equation}w = \Big(\frac{z-1}{z+1}\Big)^{2} \end{equation}[/tex]
    on some domain which contains [tex]z=e^{i\theta}[/tex]. [tex]\theta[/tex] between 0 and [tex]\pi[/tex]
    Hint: Map the semicircular arc bounding the top of the disc by putting [tex]$z=e^{i\theta}$[/tex] in the above formula. The resulting expression reduces to a simple trig function.

    2. Relevant equations
    I can get the map if I can figure out what function they're going for, but I have no idea what function this is.


    3. The attempt at a solution
    [tex]$$w = \Big(\frac{e^{i\theta}-1}{e^{i\theta}+1}\Big)^{2}$$[/tex]
    Where the heck do I go from here?
     
  2. jcsd
  3. Apr 19, 2010 #2

    tiny-tim

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    Welcome to PF!

    Hi EnginerdRuns! Welcome to PF! :smile:

    Multiply top and bottom of the fraction by e-iθ/2 :wink:
     
  4. Apr 19, 2010 #3
    Thanks bro. I'm running off of way too little sleep at this point.
     
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